Gaps in the ordinals writable by Ordinal Turing Machines with a single countable parameter

Let $$W(\alpha)$$ denote the set of all (countable) ordinals writable by Ordinal Turing Machines with a single (countable) parameter $$\alpha$$, i.e. each computation starts with a single ($$\alpha$$-th) cell on the input tape marked with a non-zero symbol (all other cells are marked with a zero symbol).

Question: does there exist an ordinal $$\beta$$ such that there exists at least one ordinal $$\gamma < \beta$$ such that $$\gamma \notin W(\beta)?$$ If no, why? If yes, how large is the smallest such $$\beta$$?

The answer is yes. For example, take $$\beta=\omega_1$$, the first uncountable ordinal. Since there are only countably many programs, there can be only countably many writable ordinals relative to $$\beta$$ as input. But there are uncountably many ordinals $$\gamma$$ below $$\beta$$, and so most of them are not writable from $$\beta$$.
The smallest such ordinal will be exactly $$\omega_1^L$$. To see this, consider any $$\beta<\omega_1^L$$. Thus, $$\beta$$ is countable in $$L$$, and this will be revealed in some countable stage of the constructibility hierarchy $$L_\eta$$. Consider the algorithm that on input $$\beta$$ begins to construct copies of the constructible hierarchy until it finds a stage $$L_\eta$$ that can see that $$\beta$$ is countable. Thus, this structure provides an $$\omega$$-enumeration of the ordinals $$\gamma<\beta$$. Therefore they will all be writable from $$\beta$$, since there will be a program that searches for this $$L_\eta$$ and then writes the $$n$$th such $$\gamma$$ in that enumeration.